Theorem sbal1yz 1989

Description: Lemma for proving sbal1 1990. Same as sbal1 1990 but with an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 23-Feb-2018.)

Assertion

Ref	Expression
sbal1yz	⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal1yz

Step	Hyp	Ref	Expression
1		dveeq2or 1804	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧)
2		equcom 1694	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
3	2	nfbii 1461	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 ↔ Ⅎ𝑥 𝑧 = 𝑦)
4		19.21t 1570	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
5	3, 4	sylbi 120	. . . . . . 7 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
6	5	orim2i 751	. . . . . 6 ⊢ ((∀𝑥 𝑥 = 𝑧 ∨ Ⅎ𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))))
7	1, 6	ax-mp 5	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
8	7	ori 713	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑)))
9	8	albidv 1812	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑)))
10		alcom 1466	. . . 4 ⊢ (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑧 = 𝑦 → 𝜑))
11		sb6 1874	. . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑))
12	2	imbi1i 237	. . . . . . 7 ⊢ ((𝑦 = 𝑧 → 𝜑) ↔ (𝑧 = 𝑦 → 𝜑))
13	12	albii 1458	. . . . . 6 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → 𝜑))
14	11, 13	bitri 183	. . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑧 = 𝑦 → 𝜑))
15	14	albii 1458	. . . 4 ⊢ (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑧 = 𝑦 → 𝜑))
16	10, 15	bitr4i 186	. . 3 ⊢ (∀𝑦∀𝑥(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
17		sb6 1874	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑))
18	2	imbi1i 237	. . . . 5 ⊢ ((𝑦 = 𝑧 → ∀𝑥𝜑) ↔ (𝑧 = 𝑦 → ∀𝑥𝜑))
19	18	albii 1458	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) ↔ ∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑))
20	17, 19	bitr2i 184	. . 3 ⊢ (∀𝑦(𝑧 = 𝑦 → ∀𝑥𝜑) ↔ [𝑧 / 𝑦]∀𝑥𝜑)
21	9, 16, 20	3bitr3g 221	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
22	21	bicomd 140	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698  ∀wal 1341  Ⅎwnf 1448  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbal1  1990